Multiple-input multiple-output (MIMO) systems can provide increased reliability in wireless communication links by exploiting the spatial diversity due to the increased number of transmit-receive paths. A simple technique to obtain the highest possible diversity order is to employ transmit beamforming and receive combining, which also improves the array gain. This technique requires that the transmitter has channel state information in the form of a transmit beamforming vector. It is often impractical to have a full reciprocal channel from receiver to transmitter to enable the transmitter to estimate the forward channel state information. Instead, the receiver estimates the channel state information, computes the corresponding beamforming vector, and encodes the beamforming vector in a small number of bits. These bits are sent via a feedback path to enable the transmitter to generate the beamforming vector. Such systems are known as limited feedback systems.
The most straightforward approach to designing a limited feedback system is to employ scalar quantization, where each component of the beamforming vector is quantized and encoded separately. In more advanced limited feedback systems, the transmitter and receiver share a codebook of possible beamforming vectors indexed by a number of bits. The receiver chooses a beamforming vector from the codebook on the basis of maximizing the effective signal-to-noise ratio (SNR) after combining, and sends the corresponding bits to the transmitter.
Beamforming vector codebooks are conventionally designed using the minimum number of feedback bits possible for a given effective SNR after combining, i.e. neglecting the search and storage requirements for the codebook. Codebook design strategies generally use numerical optimization techniques, or for larger systems, the codebooks can be randomly generated (i.e. random vector quantization, or RVQ). Such random codebooks have been shown to be asymptotically optimal as the number of bits and transmit antennas increase.
Unfortunately, the codebook size increases exponentially with the number of transmit antennas to maintain a given effective SNR or capacity loss with respect to the optimal unquantized system. Since RVQ codebooks have no structure, an exhaustive search is usually required to find the bits encoding a given beamforming vector, or vice versa. For time-varying channels, the resulting delay due to the excessive search time reduces the effectiveness of the beamforming vector when employed at the transmitter. The computation required for such an exhaustive search also consumes power, which is undesirable for low-power mobile wireless devices. Non-exhaustive methods for searching unstructured codebooks at the expense of increased memory requirements have been documented. One of these methods is a tree-search, where storage of the tree and codebook is required. An additional consequence of the exponential growth in codebook size with antenna number is that storage of the codebook may be infeasible for large numbers of antennas.
The problem of codebook search time and storage requirements is of particular importance to multiuser systems, where quantization errors increase the interference between users.